Kainui
  • Kainui
For any permutation matrix P, show that every matrix can be expressed as the sum of a matrix that commutes with P and another matrix that anticommutes with P.
Mathematics
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SOLVED
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katieb
  • katieb
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Kainui
  • Kainui
Solution: A is an arbitrary matrix: A = B + C Where BP=PB and CP=-PC are the relations P is a permutation matrix so it's its own inverse: \(PP=I\) Now I show that these definitions of B satisfies this relation: \[B=\frac{A+PAP}{2}\] \[BP = \frac{(A+PAP)P}{2} = \frac{AP+PA}{2} = \frac{P(A+PAP)}{2} = PB\] The rest is pretty straight forward to show on your own, but feel free to ask questions if you have 'em!
Kainui
  • Kainui
This is what I do when I'm bored lol

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